Application of Inversion of a Matrix in Cryptography - International ...

B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
A NOTE ON THE APPLICATION OF QUADRATIC FORMS IN
CODING THEORY WITH A NOTE ON SECURITY.
B
1
2
3
. Vellaikannan
Dr.
V.
Mohan
V.
Gnanaraj
1. Senior Grade Lecturer in Mathematics, E Mail : bvkmat@tce.edu, 9944563124, 0452-2562693
Thiagarajar College of Engineering, Madurai - 625 015. Tamil Nadu , India.
2. Professor and HOD of Mathematics, E Mail : vmohan@tce.edu, 9894026923, 0452-2482240
Thiagarajar College of Engineering, Madurai - 625 015, Tamil Nadu , India.
3. Selection Grade Lecturer in Mathematics, E Mail : vgmat@tce.edu 9442784592, 0452-2482240
Thiagarajar College of Engineering, Madurai - 625 015. Tamil Nadu, India.
ABSTRACT
A novel approach which incorporates the salient features of message sharing
(Called coding theory) is presented and also extended to the messages of higher
length. The proposed method is very simple in its principle and has great potential
to be applied to other situations where the exchange of messages is done
confidentially.
Keywords
Quadratic forms – Canonical form -- Matrices – Matrix Multiplication –
Inverse Matrix – Invertible Matrices – Diagonal matrices – Matrix induced by
a Quadratic form -Encoder –Decoder – Message Matrix.
1. Introduction
Coding theory is a subject by which the exchange of messages is
administered in a confidential and more secured way having a wide application in Military
operations, Banking Transactions etc… Recently there has been a wide application of
inversion of matrices (of order 2 whose inverses are readily obtained) [1], [2] to the
problem of exchange of messages in a confidential and a secured way. In the proposed
method the idea has been extended to the matrices mainly to nonsingular diagonal
matrices of higher order, especially induced from Quadratic forms. We know that
determination of the inverse of nonsingular matrices of higher order is difficult and this
requires higher level algorithms for the use of computers.
A newly developed method that avoids the difficulties in the
determination of inverse of a nonsingular matrix is introduced in this paper. The results
obtained using this method works vey well for the whole range of message exchanging
problems and the excellent agreement with the existing one.
1.1 Theorem
A text message of strings of some length / size l can be converted in to a matrix
(called a message matrix M) of size m × n where
n < m and n is the least such that mx n ≥ l depending up on the length of the
message with help of suitably chosen numerals and zeros.
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
Proof :
The proof is by enumeration on numbers.
Consider the following
For the text message of length up to l = 9 ; then we have m = 3 ; n = 3.
Similarly For the text message of length up to l = 12 we have m = 4 ; n = 3.
And so on. Hence the proof.
2. Basics
Consider the text message
I LOVE VINODHINI
To every letter we will associate a number. The easiest way to do that is to associate 0 to
a blank or space, 1 to A, 2 to B, etc... Another way is to associate 0 to a blank or space, 1
to A, -1 to B, 2 to C, -2 to D, etc... Let us use the second choice. So our message is given
by the string as
I L O V E V I N O D H I N I
5 0 -6 8 -11 3 0 -11 5 -7 8 -2 -4 5 -7 5
Now we rearrange these numbers into a matrix M (Row wise e/ Coolum wise).we use
column wise. For our case, we have
⎛ 5
⎜
⎜ 0
⎜ − 6
M = ⎜
⎜ 8
⎜
⎜
−11
⎝ 3
0
−11
5
− 7
8
− 2
− 4⎞
⎟
5 ⎟
− 7⎟
⎟
5 ⎟
0
⎟
⎟
0
⎠
of order ( 6 × 3)
(Using Theorem)
Then we perform the product M A , where A is an arbitrary nonsingular matrix given by
⎛ −1
5
⎜
A = ⎜−
2 11
⎜
⎝ 1 − 5
Then, we get
−1⎞
⎟
7 ⎟
2 ⎟
⎠
−1
whose inverse is given by A =
⎛57
⎜
⎜ 11
⎜
⎝−1
− 5
−1
0
46⎞
⎟
9 ⎟
−1⎟
⎠
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
X = MA=
⎛ 5
⎜
⎜ 0
⎜ − 6
⎜
⎜ 8
⎜
⎜
−11
⎝ 3
0
−11
5
− 7
8
− 2
− 4⎞
⎟
5 ⎟
⎟ ⎛ −1
− 7 ⎜
⎟ ⎜−
2
5 ⎟ ⎜
0
⎟ ⎝ 1
⎟
0
⎠
5
11
− 5
−1⎞
⎟
7 ⎟
2 ⎟
⎠
=
⎛ − 9
⎜
⎜ 27
⎜−11
⎜
⎜ 11
⎜
⎜
− 5
⎝ 1
45
−146
60
− 62
33
− 7
−13
⎞
⎟
− 67⎟
27 ⎟
⎟
− 47⎟
67
⎟
⎟
−17
⎠
The encoded numeric message to be sent is
-9, -27, -11, 11,………-47,67,-17.
This encoded message is again decoded using the inverse of A as
M =
−1
XA =
⎛ − 9
⎜
⎜ 27
⎜−11
⎜
⎜ 11
⎜
⎜
− 5
⎝ 1
45
−146
60
− 62
33
− 7
−13
⎞
⎟
− 67⎟
⎟ ⎛57
27 ⎜
⎟ ⎜ 11
− 47⎟
⎜
67
⎟ ⎝−1
⎟
−17
⎠
− 5
−1
0
46⎞
⎟
9 ⎟
−1⎟
⎠
⎛ 5
⎜
⎜ 0
⎜ − 6
= ⎜
⎜ 8
⎜
⎜
−11
⎝ 3
0
−11
5
− 7
8
− 2
− 4⎞
⎟
5 ⎟
− 7⎟
⎟
5 ⎟
0
⎟
⎟
0
⎠
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
This matrix is again converted in to a string of numerals as
5 0 -6 8 -11 3 0 -11 5 -7 8 -2 -4 5 -7 5
I L O V E V I N O D H I N I
3. Solution Procedure
A text message of strings of some length / size l from the sender is converted in
to a stream of numerals with the help of some coding process (Probably may be the
standard codes like A – 1, B-2,….Z-26 and for space -0 ) which is again converted in to
a matrix (called a message matrix M) of size m × n where
n < m and n is the least such that mx n ≥ l depending up on the length of the
message .In such case the size of the Encoder (The induced Diagonal matrix of a
Quadratic form of suitable variables ) becomes n . Then the Encoder need not be an
arbitrary matrix where as it may be taken as a Diagonal matrix of size n whose inverse
can be readily obtained.
Then the message matrix is converted in to a New Matrix X (Encoded Matrix)
using Matrix Multiplication as X = ME . Then this is sent to the Receiver. Then the
receiver decode this matrix with the help of a matrix D (Decoder matrix) which is
−1
nothing but the inverse of the encoder ( i . e.,
D = E ) , to get the message matrix back as
−1
M = XE .Then with the previously used codes the receiver can get back the message
in terms of the numerals which again can be converted to the original text message. When
the length / size of the text message is too large, the value of n become higher, leading
to the need of higher order diagonal matrices induced from the quadratic forms of higher
number of variables.
3. 1 Algorithm
3.1.1 Encoding Process
1. Convert the text message of length l in to a stream of Numerals using a user
friendly scheme for both the .sender and the receiver.
2. Place the numerals in to a matrix of order m × n where
n < m and n is the least such that mx n ≥ l where n depends on the size
of the message and call this as a Message matrix M.
3. Multiply this message matrix by the Encoder E of size n. (Normally a induced
diagonal matrix compatible for the product X = ME .) and get the encoded
matrix X.
4. Convert the message matrix in to the stream of numbers that contains the
encrypted message and sent to the receiver.
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
3.1 .2 Decoding Process
1. Place the encrypted stream of numbers that represent the encrypted message
in to a matrix
−1
2. Multiply the encoded matrix X with the decoder D = E (The inverse of E) to
get back the message matrix M
3. Convert this message matrix in to a stream of numbers with the help of the
originally used scheme.
4. Convert this stream of numerals in to the text of the original message.
4. Results and Discussions
Consider the message to be sent: BEST WISHES
We take the standard codes as follows:
A →1 ; B → 2 ; ........; Z → 26 and Space → 0
1. We convert the above message in to a stream of numerical values as
follows:
BEST WISHES 2 5 19 20 0 23 9 19 8 5 19 (Here we admit a single
spacing for the purpose better understanding)
2. We construct the message matrix M with this stream of numerals as
⎛ 2 5 19⎞
⎜ ⎟
⎜20
0 23⎟
M = ⎜ ⎟ which is of order 4x 3.
(Using Theorem)
9 19 8
⎜ ⎟
⎝ 5 19 0 ⎠
3. Based on this, We take the 3 rd order Diagonal matrix (The
diagonalized matrix of the matrix of a QF of suitable variables
otherwise called the matrix of the canonical form) with Diag ( )).
2 2 2
For e:g if the QF is 2x1 + x2
+ x3
+ 2x1x2
− 2x1x3
− 4x2
x3
then the
matrix of the QF is
⎛ 2 1 −1⎞
⎜
⎟
2 2 2
⎜ 1 1 − 2⎟
.Also the canonical form is − y
1
+ y2
+ 4y3
whose
⎜
⎟
⎝−1
− 2 1 ⎠
⎛−1
0 0⎞
⎜ ⎟
matrix is given by D ( −1,1,
4 ) = ⎜ 0 1 0⎟
⎜ ⎟
⎝ 0 0 4⎠
⎛−1
0 0⎞
⎜ ⎟
4. Then we have the Encoder as E = ⎜ 0 1 0⎟
.
⎜ ⎟
⎝ 0 0 4⎠
5. Then the encoded matrix is given by
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
⎛ 2 5 19⎞
⎛ − 2 5 76⎞
⎜ ⎟ ⎛−1
0 0⎞
⎜
⎟
⎜20
0 23⎟
⎜ ⎟ ⎜−
20 0 92⎟
X = ME = ⎜ ⎟ ⎜ 0 1 0⎟
= ⎜ − ⎟ .
9 19 8
⎜ ⎟ ⎜ ⎟ 9 19 32
⎝ ⎠
⎜
⎟
0 0 4
⎝ 5 19 0 ⎠
⎝ − 5 19 0 ⎠
Hence the encoded numeric message is given by
-2 5 76 -20 0 92 -9 19 32 -5 19 0
⎛ ⎞
⎜ = 1 0 0 ⎟
−1
−1
6. Clearly the Decoder E is the given by E =
⎜
0 1 0
⎟
.
⎜ 1 ⎟
⎜ 0 0 ⎟
⎝ 4 ⎠
7. The encoded numeric message is to be decoded by first writing the
encoded matrix X from the received message as
⎛ − 2 5 76⎞
⎛ ⎞ ⎛ 2 5 19⎞
⎜
⎟ ⎜ = 1 0 0 ⎟ ⎜ ⎟
−1
⎜−
20 0 92⎟
⎜ ⎟ ⎜20
0 23⎟
M = XE = ⎜
⎟
0 1 0 =
−
⎜ ⎟ ⎜ 9 19 8 ⎟ .
9 19 32
⎜
⎟
1
⎜ ⎟ ⎜ ⎟
0 0
⎝ − 5 19 0 ⎠ ⎝ 4 ⎠ ⎝ 5 19 0 ⎠
8. This matrix M is converted in to numeric message as
2 5 19 20 0 23 9 19 8 5 19
9. This stream of numerals is converted in to the text message as
2 5 19 20 0 23 9 19 8 5 19: BEST WISHES
4. 1 A word on Security:
In case of using the standard codes one could recognize intuitively
or by any way the codes of use from the codes allotted for the alphabets. So the use of
codes in a random or chaotic way or by using some process , increases the security level.
4.1.1 Example
Instead of using the standard codes A-1, B-2…….Z-26 and 0 for space
If we use the codes assigned as
A-7 , B-6 , C- 5 , D-4 , E-3 ,F-2 ,G-1 , H-8 , I-9 , J-10 , K-11 , L-12 , M-13 . N- 15 , O-
16, P-17 , Q- 18 , R- 19, S-14 , T-20 , U-26 , V-25 , W-24 , X- 23 , Y -22 ,Z- 21. and 0
for space.(In a random way or by using some generator using Number theory or
combinatorics)
Then the message BEST WISHES is given by the matrix
⎛ 6 3 14 ⎞
⎜
⎟
⎜20
0 24⎟
M = ⎜
⎟ Instead of
9 14 8
⎜
⎟
⎝ 3 114 0 ⎠
M
⎛ 2
⎜
⎜20
= ⎜ 9
⎜
⎝ 5
5
0
19
19
19⎞
⎟
23⎟
8 ⎟
⎟
0
⎠
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
Any one who intervene the communication uses the standard codes for this
message matrix will get a confusing message like FENT XINHCN.
So the messengers are advised to make use of their convenient system of codes in order
to have higher security level.
5. Operation on Strings:
We define the operator * (The string addition) as usual in the case
of addition of strings.
Example: Best*wishes = Best wishes.
6. Generalization
Using this operation we decompose the messages of larger length in to
messages of shorter lengths and finally these are coined to get the message of larger
length.
6.1 Results and Discussion
Consider the Message M: MEPCO WISHES YOU ALL THE BEST.
This message is decomposed in to two messages as follows,
M = M1 + M2 Where M1 = MEPCO WISHES & M2 = YOU ALL THE BEST.
Now for M1:
1) M E P C O W I S H E S:
13 5 16 3 15 0 23 9 19 8 5 19
⎛13
5 16⎞
⎜ ⎟
⎜ 3 15 0 ⎟
2) T
1
= ⎜ ⎟ .
23 9 19
⎜ ⎟
⎝ 8 5 19⎠
3)
⎛= 1 0 0⎞
⎜ ⎟
E = ⎜ 0 1 0⎟
Such that
⎜ ⎟
⎝ 0 0 4⎠
4) = T E =
X1
1
T
1
⎛ −13
⎜
⎜ − 3
= ⎜−
23
⎜
⎝ − 8
5
15
9
5
64⎞
⎟
0 ⎟
76⎟
⎟
76
⎠
E
−1
⎛
⎜ = 1
=
⎜
0
⎜
⎜ 0
⎝
0
1
0
⎞
0 ⎟
0
⎟
1
⎟
⎟
4 ⎠
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
−1
5) M = X =
1 1E
⎛13
⎜
⎜ 3
⎜23
⎜
⎝ 8
5
15
9
5
16⎞
⎟
0 ⎟
19⎟
⎟
19
⎠
6) Message 1 = 13 5 16 3 15 0 23 9 19 8 5 19
M E P C O W I S H E S
Now for M2:
1) Y O U A L L T H E B E S T.
25 15 21 0 1 12 12 0 24 8 5 0 2 5 19 20
2)
3)
⎛25
15 21⎞
⎜ ⎟
⎜ 0 1 12⎟
⎜12
0 21⎟
T = ⎜ ⎟
2
.
⎜ 8 5 0 ⎟
⎜ ⎟
⎜
2 5 19
⎟
⎝20
0 0 ⎠
⎛= 1 0 0⎞
⎜ ⎟
E = ⎜ 0 1 0⎟
Such that
⎜ ⎟
⎝ 0 0 4⎠
⎛−
25
⎜
⎜ 0
⎜ −12
4) X
2
= T2
E = ⎜
⎜ − 8
⎜
⎜
− 2
⎝−
20
15
1
0
5
5
0
84⎞
⎟
48⎟
84⎟
⎟ ,
0 ⎟
76
⎟
⎟
0
⎠
E
−1
⎛
⎜ = 1
=
⎜
0
⎜
⎜ 0
⎝
0
1
0
⎞
0 ⎟
0
⎟
1
⎟
⎟
4 ⎠
−1
5) M = X =
2 2
E
⎛25
⎜
⎜ 0
⎜12
⎜
⎜ 8
⎜
⎜
2
⎝20
15
1
0
5
5
0
21⎞
⎟
12⎟
21⎟
⎟
0 ⎟
19
⎟
⎟
0
⎠
6) Message 2 = 25 15 21 0 1 12 12 0 24 8 5 0 2 5 19 20
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B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
Y O U A L L T H E B E S T.
Therefore the message M = M1 * M2
= M E P C O W I S H E S YOU A L L T H E B E S T.
6. Results Used from matrix theory
The following are the results used in our paper from the theory of
matrices available in [3]
1. To any Quadratic there exists a matrix called the Matrix of quadratic
form.
2. The matrix of any Quadratic form is a Real Symmetric matrix.
3. The Eigen values of a real symmetric matrix are always real.
4. Any Quadratic form can be reduced to canonical form by means of
orthogonal reduction.
5 The matrix of the canonical form of a Quadratic form is a diagonal
matrix.
6. The inverse of a diagonal matrix with a
ii
as entries is nothing but the
7. Conclusion
scalar diagonal matrix with
1 as entries.
a ii
1. Diagonal matrices induced from Quadratic forms are preferred for
encoding as their inverses can be easily obtained.
2. This provides a transaction of least amount of messaging between the
sender and the receiver. (It is sufficient to know the codes of use and the
Quadratic form).Here the security is assured as only those know about the
Quadratic Forms can understand the process.
3. Higher order Diagonal matrices are preferred as their inverses are easily
found.
4. When the size of the message is too large new string operations may be
defined and the message can be splitted and suitable such processing may
be carried over.
5. Higher level of security can be achieved by using own conventional
codes or codes (As in the word on Security) processed by some structure.
8. Scope in further work
1. Texts with all types of characters may be utilized in the study.
2. Higher level of security can be enhanced by using structured system of
codes.
3. Search of Diagonal matrices induced from any other field may be done.
4. Efforts can be taken for the use of any non singular matrix as an Encoder.
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9. Acknowledgements
The authors are very grateful to Major D.Stephen Jeyapaul , Former Head , Dept
of Mathematics , Thiagarajar College of Engineering , Prof.K.P.Radhakrishnan Former
Head , Dept of Mathematics , Thiagarajar College of Arts Madurai , for their constant
encouragement. The author sincerely thanks Dr.P.S.Boopathi Manickam, Dept of
Humanities and Mr.S.Rajaram, Dept of English for their scholarly suggestions and timely
help. Above all the author dedicates this work to his ever loving family members
particularly to his parents without whom he can’t succeed up to this level.
10. References
B.Vellaikannan,Dr.V.Mohan,V.Gnanaraj.,Int. J. Comp.Tech. Appl,Vol 1 (1), 78-87
[1]. http://www. richland.edu / james /lecture /.../matrices/applications.html
[2]. http:// aix1.uottawa.ca/~jkhoury/cryptography.htm
[3]. Vasta B.S., Vasta Suchi..,Theory of Matrices.,Third edition., New Age
International , India., 2010.
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